ON SECOND EIGENVALUES OF CLOSED HYPERBOLIC SURFACES FOR LARGE GENUS

Abstract

In this article, we study the second eigenvalues of closed hyperbolic surfaces for large genus. We show that for every closed hyperbolic surface Xg of genus g (g ≥ 3), up to uniform positive constants multiplications, the second eigenvalue λ2(Xg) of Xg is greater than L2(Xg)/g2 and less than L2(Xg); moreover these two bounds are optimal as g → ∞. Here L2(Xg) is the shortest length of simple closed multi-geodesics separating Xg into three components. Furthermore, we also investigate the quantity λ2(Xg)/L2(Xg) for random hyperbolic surfaces of large genus. We show that as g → ∞, a generic hyperbolic surface Xg has λ2(Xg)/L2(Xg) uniformly comparable to 1/ln(g).